\(L^p\) boundedness for singular integrals with rough kernels on product domains
In this paper, the authors consider the singular integral operator \(T(f)\) defined by \[ T(f)(x,y)= \text{p.v. }\int_{\mathbb{R}^n} \int_{\mathbb{R}^m} h(|u|,|v|) \Omega(u,v)|u|^{-n}|v|^{-m} f(x-{\mathcal P}(u), y-{\mathcal Q}(v)) du dv, \] where \(\Omega\) is an integrable function on the product unit spheres \(S^{n-1}\times S^{m-1}\), that satisfies the following conditions: \[ \Omega(tx, sy)= \Omega(x,y),\quad\text{for some }s,t\in \mathbb{R}^+; \] \[ \int_{S^{n-1}} \Omega(x,y) d\sigma(x)= \int_{S^{m-1}}\Omega(x,y) d\sigma(y)= 0\quad\text{for any }y\in S^{m-1},\;x\in S^{n-1}; \] the function \(h\) satisfies \[ \sup_{\alpha,\beta> 0} \alpha^{-1} \beta^{-1}\int^\alpha_0 \int^\beta_0|h(s,t)|^\nu ds dt< \infty \] for some \(\nu> 1\); and \({\mathcal P}= (P_1,\dots, P_N)\), \({\mathcal Q}= (Q_1,\dots, Q_M)\) are two real polynomial mappings from \(\mathbb{R}^n\) to \(\mathbb{R}^N\) and \(\mathbb{R}^m\) to \(\mathbb{R}^M\), respectively. Let \(B^{0,1}_q(S^{n-1}\times S^{m-1})\) be a block space for some \(q> 1\). The authors prove the following result. Theorem A. Suppose that \(\Omega\in B^{0,1}_q\) for some \(q> 1\). (i) For \(|1/p- 1/2|<\min\{1/2, 1/\nu'\}\), \(T\) is bounded on \(L^p(\mathbb{R}^n\times \mathbb{R}^M)\). (ii) In addition, if \({\mathcal P}(-u)= -{\mathcal P}(u)\) and \({\mathcal Q}(-v)= -{\mathcal Q}(v)\) then \(T\) is bounded on all \(p\in (1,\infty)\). The bounds on (i) and (ii) are independent of the coefficients of \(\{P_i\}\) and \(\{Q_j\}\). Moreover, assuming \(h(s,t)= h_1(s) h_2(t)\), the authors obtain the same boundedness results for the truncated maximal singular integrals.
scientific article; zbMATH DE number 1876750